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Abstract

Using the nonequilibrium Green’s function method, we theoretically study the Andreev
reflection(AR) in a four-terminal Aharonov-Bohm interferometer containing a coupled
double quantum dot with the Rashba spin-orbit interaction (RSOI) and the coherent
indirect coupling via two ferromagnetic leads. When two ferromagnetic electrodes are
in the parallel configuration, the spin-up conductance is equal to the spin-down conductance
due to the absence of the RSOI. However, for the antiparallel alignment, the spin-polarized
AR occurs resulting from the crossed AR (CAR) and the RSOI. The effects of the coherent
indirect coupling, RSOI, and magnetic flux on the Andreev-reflected tunneling magnetoresistance
are analyzed at length. The spin-related current is calculated, and a distinct swap
effect emerges. Furthermore, the pure spin current can be generated due to the CAR
when two ferromagnets become two half metals. It is found that the strong RSOI and
the large indirect coupling are in favor of the CAR and the production of the strong
spin current. The properties of the spin-related current are tunable in terms of the
external parameters. Our results offer new ways to manipulate the spin-dependent transport.

Keywords:

Background

A quantum dot (QD) is an artificially low-dimensional structure that can be filled
with electrons (or holes). Two or more QDs can be coupled to form multiple-QD systems
(i.e., artificial molecules). Because the degrees of freedom of the QDs are well controllable,
it is possible to add or remove the electrons in the QDs, and the QD system can be
coupled via tunnel barriers to electrodes, in which electrons can be exchanged. Accordingly,
the artificial molecule provides an excellent model system in which the thorough investigation
of quantum many-body properties in a confined geometry can be performed
[1-6]. Among the various multiple-QD systems, an Aharonov-Bohm (AB) interferometer containing
double QDs (DQDs) is of particular interest and importance, in which two QDs are embedded
in the opposite arms of the AB ring, respectively, and they are coupled to each other
via barrier tunneling. As a tunable two-level system, the parallel DQD system that
can become one of the promising candidates for the quantum bit in quantum computation
has received more attention
[7-20]. However, in an actual DQD system, the coherent indirect coupling between two QDs
via a reservoir is very essential. Kubo et al. introduced the parameter 1αcharacterizing the indirect coupling strength, and Gurvitz also indicated the fundamentality
of the sign of the coherent indirect coupling parameter
[21,22]. Kubo et al. investigated the pseudospin Kondo effect in a lateral DQD system using
the slave-boson mean-field method and found that the exotic pseudospin Kondo effect
occurs when a coherent indirect coupling is presented through the common reservoirs
[23]. Recently, Kubo and co-workers calculated the shot noise and Kondo effect in a DQD
structure with the coherent indirect coupling. Their results demonstrate that the
coherent indirect coupling can generate a novel antiferromagnetic exchange phenomenon
[24]. Trocha and Barnaś studied theoretically the spin-dependent transport through a DQD
coupled to ferromagnetic leads. They observed that the Fano antiresonance of the linear
conductance relies on the sign of the indirect coupling in the nondiagonal coupling
elements
[8]. Furthermore, the transport properties of a DQD system has been considered in the
orbital Kondo regime. That the Kondo temperature and Kondo resonances are susceptible
to the coherent indirect coupling parameter is also revealed
[25]. In addition, if a QD is formed in a semiconductor two-dimensional electron gas structure
without the inversion symmetry in the growth direction, the Rashba spin-orbit interaction
(RSOI) will emerge, and the RSOI can induce the spin-related phase factor in the tunneling
matrix elements and the spin-flip effect. The RSOI results from a relativistic effect
at the low speed limit, and it can couple the electron spin to its orbital motion,
thus providing a possible way to control the spin degree of freedom by means of an
external electric field. As a consequence, the coherent indirect coupling and the
RSOI make the quantum transport through the QD systems rich and varied
[26-30].

On the other side, the subgap transport through heterostructures with nano-objects
(such as QDs, molecules, nanowires, etc.) coupled to one conductor and another superconducting
lead has attracted a great deal of attention over the past years due to the fundamental
physics and its potential applications
[31-35]. Andreev reflection (AR) usually occurs in the hybrid systems, in which two electrons
with opposite spins enter the superconductor from the normal metal region, leading
to the formation of a Cooper pair in the superconducting region
[36-38]. In comparison with the standard mechanism of normal AR, the crossed AR (CAR) is
a nonlocal dynamics process which occurs at the contact between a superconductor and
two normal leads, where two subgap electrons from different metals enter into the
superconductor and generate a Cooper pair there
[39-42]. AR (or CAR) in nanoscopic heterostructures gives rise to a rich subgap structure
in the current-voltage characteristics. Accordingly, understanding the AR and CAR
has attracted theoretical and experimental attention mainly because the AR (or CAR)
may create the entangled electrons in a solid-state device, and CAR can be readily
probed by spin selection using ferromagnetic electrodes. This approach is almost unrealized
for entangler devices, since projecting the spin will cause the destruction of entanglement
[43]. Based on the CAR, the controlled Cooper pair splitting has been realized in terms
of a two-quantum dot Y-junction
[44], which opens a possible route towards a test of the Einstein-Podolsky-Rosen (EPR)
paradox and Bell inequalities in solid-state systems. Herrmann et al. used carbon
nanotube DQD as Cooper pair beam splitters and realized the quantum optic-like experiments
with spin-entangled electrons
[45]. These results show that the CAR has an important application in testing a fundamental
property of quantum mechanics.

To our knowledge, the AR in the DQD with a maximum coupling |α| = 1 has been studied widely. However, the quantum transport through a four-terminal
AB interferometer including a DQD in the presence of the AR, the coherent indirect
coupling, and RSOI is less explored. Motivated by recent theoretical and experimental
advances in the DQD systems
[7-10,13,15,16,19,21-25,44,45], one may expect that the interplay of the coherent indirect coupling and the RSOI
in the presence of the AR will add new physics to hybrid quantum systems, which may
have practical applications for future spintronics. Consequently, we investigate the
AR in the above-mentioned system in this paper. It is found that the RSOI and a nonzero
coherent indirect coupling cause the spin-polarized AR when the polarizations of two
ferromagnetic leads are parallel, but for antiparallel (AP) arrangement of the polarizations
of two ferromagnetic leads, the CAR can contribute the spin-polarized AR conductance.
We note that the convex shape of the Andreev-reflected tunneling magnetoresistance
(ARTMR) versus the magnetic flux relies on the sign of the coherent indirect coupling
parameter, and there are extreme values in the plot of the ARTMR versus the coherent
indirect coupling parameter. Even the negative ARTMR also occurs. This is a spin valve
effect in the AR process. It is interesting to note that the sign of the coherent
indirect coupling parameter leads to the swap effect in the spin-polarized current
plot, and the pure spin current can be produced when two ferromagnetic leads are fully
polarized. The spin-dependent AR current can be controlled by means of the gate voltage,
RSOI, magnetic flux, and so on. These results provide the ways to manipulate the spin-dependent
transport by means of the system parameters.

Methods

We consider a hybrid four-terminal AB interferometer including a parallel DQD coupled
to two ferromagnetic reservoirs and two superconductors as shown in Figure
1. The system is described by the following Hamiltonian:

where HF is the Hamiltonian of the left and right ferromagnetic electrodes

(2)

Here,
is the creation (annihilation) operator in the lead νwith energy εν,kσ. HS represents two superconducting reservoirs with chemical potential μs = 0 and the energy gap Δ,

(3)

HDQD in Equation 1 denotes the DQD Hamiltonian

(4)

in which
represents the creation (annihilation) operator of the electron with energy εi in the dot i; tc is the coupling strength taken as a real parameter. The last term, HT, in Equation 1 corresponds to the tunneling Hamiltonian between the DQD and four
leads,

(5)

where the tunneling matrix elements between the DQD and two ferromagnetic leads are
,
,
, and
. The phase shift due to the total magnetic flux threading into the AB ring is assumed
to be ϕ = 2Π(ΦL + ΦR)/ϕ0 with the flux quantum ϕ0 = h/e. The phase factor φRi comes from the RSOI in dot i, which is tunable in the experiments
[46,47].
as the tunneling coupling between the DQD and two superconductors is also assumed
to be independent of k and σ.

Using the nonequilibrium Green’s function technique, the spin-dependent current through
the left ferromagnetic reservoir can be expressed as
[48,49]

(6)

where Tr is the trace in the spin space;
is a 4 × 4 matrix with Pauli matrix σz as its diagonal components; Gr,a,<(ε) are retarded, advanced, and lesser Green’s functions in the generalized 4 × 4 Nambu
notation.

(7)

(8)

with the vector
.

After some algebraic manipulations, the spin-dependent current can be derived from
Equation 6:

(9)

(10)

in which
and
are the spin-dependent AR and CAR coefficients, respectively.
represents the single-particle tunneling through FL-DQD-FR or FR-DQD-FL.
corresponds to the probability of the quasiparticle tunneling among two superconductors
and the left ferromagnetic lead.
,
, and fS are Fermi-Dirac distribution functions. The derivation of the spin-dependent current
is minutely given in the Appendix.

Since we mainly focus on the AR process at zero temperature limit and set |eVL| = |eVR| < Δ,
will vanish. In the case of eVL = eVR, the current from the quasiparticle tunneling through FL-DQD-FR or FR-DQD-FL becomes
zero; as a consequence, the AR dominates the transport through the four-terminal AB
interferometer.

Results and discussions

In the following numerical calculations, we mainly elucidate the spin-dependent AR
process in the four-terminal AB interferometer with the coherent indirect coupling
and the RSOI. We take e = h = kB = 1, and set Δ = 1 as the energy unit. Throughout the paper, the symmetric couplings
with
and |PL| = |PR| are considered as a typical case.

Conductance

Because we mostly investigate the AR within the superconductor gap, in the limit of
zero bias VL = VR → 0, the spin-related AR and CAR conductances have the forms

(11)

and

(12)

It is well known that a DQD system with the maximum coupling |α| = 1 has already been investigated. Indeed, such case is very special, and most experimental
conditions correspond to |α| < 1; as a result, α characterizing the coherent indirect coupling between two QDs via two ferromagnetic
electrodes is introduced (see the Appendix). |α| < 1 comes from the various factors, such as imperfections in the ferromagnetic reservoirs
producing the destructive quantum interference, the geometrical structure of the system,
and so forth.

Let us begin with the case of ϕ = 0 and φR = Π/2; for the different coherent indirect coupling α, Figure
2 shows the total AR conductance (
and
) as a function of Fermi energy εF for parallel (P) and antiparallel (AP) configurations. In order to gain the clear
physics, the Hamiltonian HDQD is diagonalized, and two energy eigenvalues are given as
; thus, when the Fermi level coincides with the E+ and E−, the resonant AR occurs and two peaks of AR conductances are located around the level
E± as illustrated in Figure
2a, b, c. For α = 0, it is clearly seen that
is always equal to
in the P arrangement (PL = PR = 0.4), and the magnitudes of two peaks are equal. However, for the case of the AP configuration
(P L= −PR = 0.4),
appears when α = 0. Because the ferromagnetic leads have majority and minority electrons, the AR
and the CAR are governed by the minority electrons for P configuration; thus, the
AR and the CAR do not contribute the spin-polarized transport. For AP alignment, although
the AR cannot produce the spin-polarized current,
emerges due to the CAR process, in which the CAR is governed by the majority electrons.
This leads to the appearance of the spin-polarized conductance. Since two dots are
indirectly coupled via two ferromagnetic leads, which is reflected in the nondiagonal
coupled matrix elements (see Equations 15 and 16), α = 0 implies that the off-diagonal matrix elements vanish due to complete destructive
quantum interference; thus, two dots are totally decoupled through two ferromagnetic
leads. The AR (or the CAR ) can happen only through QD1 and QD2, respectively. This
leads to the conductance
for the P arrangement and the equal height of two peaks (
or
) for the AP configuration.

We also notice that both
and
occur with the increase of α for P and AP configurations; thus,
is nonzero at α ≠ 0, which means the occurrence of the spin-polarized AR for P configuration in the
presence of the RSOI and the nonzero parameter α. As a matter of fact, we have found that
is independent of the parameter α for P configuration in the absence of the RSOI, which is not shown here. In comparison
with the case of α = 0, the symmetry of AR conductances with respect to the Fermi energy is significantly
broken when α ≠ 0. It is noticeable that amplitudes of conductance peaks near the level E+ decrease, and the magnitude of the right peaks is smaller than that of the left ones.
In addition, the positions of peaks for AP alignment are also shifted with α = 1. These results indicate that the coherent indirect coupling and the RSOI play
an important role in determining the feature of the AR conductance spectra.

To elucidate better the properties of the AR under P and AP configurations, in analogy
with the conventional tunneling magnetoresistance (TMR) effect of ferromagnetic tunnel
junctions, the ARTMR is introduced and defined as

(13)

In Figure
3, we present the ϕ dependence of ARTMR for different α. The oscillation period of the ARTMR versus magnetic flux ϕ is 2Π, and the sign of the ARTMR does not change. It is interesting to note that the convex
shape of the ARTMR at ϕ = 2nΠ (n is an integer) relies on the sign of the coherent indirect coupling parameter α. In comparison to the case of |α| = 0.5, the magnitudes of ARTMR are considerably increased for |α| = 1.0. This is because the reduction of the destructive interference results in the enhancement
of ARTMR.

As we know, the RSOI can induce the spin precession and may even cause the inter-dot
spin-flip effect. According to
[26,27], the spin-dependent phase factor φR due to the RSOI can be expressed as
, where β is the RSOI strength, m∗ is the electron effective mass, and Li is the length of dot i. φR is tunable in experiments. It can reach Π/2 easily or can be larger experimentally
[27]. In order to explore further the influence of the coherent indirect coupling and
the RSOI on the ARTMR, the ARTMR as a function of the parameter α for different φR is shown in Figure
4. We can see from Figure
4 that ARTMR versus α exhibits the nonmonotonic features, and there exists the crossing point at α = 0. Since α = 0 means that the coupling off-diagonal terms in Equation 16 are totally suppressed,
as a consequence, the AMTMR is independent of the RSOI (see the Appendix). When φR is relatively small, this corresponds to the weak RSOI strength; thus, the variation
of the ARTMR with α is not smart (solid line and dashed line). However, the evolution of the ARTMR is
very remarkable as φR increases (dotted line and dash-dotted line), while the ARTMR first increases and
decreases with the increases of α, even the negative ARTMR also emerges, which corresponds to a spin valve effect in
the AR process. This reflects that the strong RSOI gives rise to the significant variation
of the ARTMR. We also observe that the maximum and minimum values appear in the curves
of the ARTMR. These demonstrate that the optimal ARTMR can be tuned by means of the
external parameters.

Spin-dependent current

Above, we analyze the properties of the AR conductances. In the following discussions,
we will explore the spin-dependent current in the AR process with the help of the
current formulas (Equations 9 and 10). To gain a full physical picture on the DQD
levels’ influence on the spin-related current, Figure
5 displays the images of the spin-polarized current Is = IL↑ − IL↓ as a function of the energy levels ε1 and ε2 of the DQD. The blue regions correspond to zero current, namely, IL↑ = IL↓ in these regimes. In the diagram, it is found that the spin-polarized current is symmetrical
about the line of ε1 = ε2 and is asymmetrical with respect to the line of ε1 = −ε2 as illustrated in Figure
5a, b. It is interesting to note that one level is aligned to the Fermi level, and
the other is far from the Fermi one (off-resonance). Is is relative small. This is because one QD is in the on-resonance state and the other
is in the off-resonance state. When both ε1 and ε2 are close to the Fermi level by tuning the gate voltage, the maximal Isappears since DQD is in the on-resonance states. We also observe that, for α = 0.5 and α = −0.5, the spin-polarized current shows the opposite feature, which is a swap effect
originating from the different sign of the parameter α. This indicates that the sign of the coherent indirect coupling parameter has a remarkable
impact on the spin-polarized current.

As we know, when ferromagnets are fully polarized, two ferromagnets become half metals
where all electrons have the same spin. AR is usually suppressed at the ferromagnet/superconductor
interface. However, AR still can occur, and the pure spin current can be generated
in the present system. For PL = −PR = 1.0 or PL = −PR = −1.0, i.e., two ferromagnetic leads become two half metals; the normal AR vanishes
due to
(see Equations 21 and 25). However, CAR dominates the transport through the four-terminal
AB interferometer for AP alignment. As a consequence, we can obtain the pure spin
current via CAR and two half-metal reservoirs. Thus, this device may be used as a
pure spin-current injector even in the absence of the RSOI. In Figure
6, we also depict AB oscillations of the spin current for different α. For the case of α = 0.5, the magnitudes of the resonant peaks and valleys are enhanced with the increase
of the RSOI strength, and positions of peaks and valleys are also shifted to the left,
as illustrated in Figure
6a, b. Since the RSOI gives rise to an extra spin-related phase factor φR (see Equation 16), the curves of the spin current versus magnetic flux ϕ move towards
the left with the emergence of the RSOI phase, and the shifted magnitude of peaks
(or valleys) is equal to φR as shown in Figure
6a, b. Physically, the increase of φR corresponds to the strong RSOI, which also favors the CAR process and the generation
of the large spin current. When the DQD is fully coupled via two ferromagnetic reservoirs
(α = 1.0), in comparison with the case of α = 0.5, it is noted from Figure
6b that not only the positions of peaks and valleys are altered, but also the amplitudes
of those are remarkably enhanced. This originates from the fact that the reduction
of the destructive interference enhances the spin current for the case of α = 1.0. These results indicate that the variation of the spin current is sensitive
to the parameter α and the strength of the RSOI, and the interplay between them determines the nature
of the spin current.

Conclusions

In this paper, we have analyzed the AR of a four-terminal AB interferometer containing
a coupled DQD with with the RSOI and the coherent indirect coupling via two ferromagnetic
leads. The formulas of the transmission coefficients are derived based on the framework
of the nonequilibrium Green’s function technique. For P configuration, the spin-polarized
AR can occur, stemming from the RSOI and a nonzero coherent indirect coupling. On
the contrary, for AP configuration, the spin-polarized AR always happens because of
the CAR mechanism. Under the introduction of the ARTMR, we find that the sign of the
ARTMR versus the magnetic flux keeps invariable for different parameter α, but the convex shape of the ARTMR depends distinctly on the sign of the parameter
α. With the increase of the RSOI strength, the ARTMR versus the parameter α exhibits the more significant nonmonotonic features, and there exist the extreme
values in the ARTMR plot, even the negative ARTMR also emerges. Since the energy levels
of the DQD can be manipulated via the gate voltage, we can obtain the optimal spin-polarized
current. A pure spin current can be generated via the CAR and two half-metal leads.
Moreover, the strong RSOI and the reduction of the destructive interference (α = 1) favor the enhancement of the spin current. Thus, this device may become an effective
spin-current generator, and the pure spin current is tuned in terms of the magnetic
flux, the RSOI strength, and so forth. These results offer the ways to manipulate
the spin-dependent transport via the four-terminal AB setup.

Appendix

In this appendix, we present the derivation of the current formulas in detail.

Let gr(ε) and Gr(ε) denote the retarded Green’s function of the DQD without and with the coupling to
the external reservoirs. In the Nambu space, gr(ε) can be given as

(14)

Based on the following Dyson equation, the retarded Green’s function of the system
can be written as Gr(ε)]−1 = gr(ε)−1−Σr, in which
. The lesser Green’s function G<(ε) = Gr(ε)Σ<Ga(ε), where Ga(ε) = Gr(ε)]† and
In the wide-band limit approximation, the retarded self-energy can be derived from
the definition

(15)

(16)

(17)

(18)

where
with ρνσ being the density of states of the spin σ band in the lead ν. We calculate the tunneling matrix element by means of the Bardeen’s formula, i.e.,
, where me is the effective mass, S is the region of the integration,
is the wave function of evanescent mode of the lead ν, and
is the wave function of an electron localized in the QD i. Considering the propagation of electrons in the reservoir ν, this propagation process (the wave number dependence of
) induces the coherent indirect coupling via the reservoir ν between two QDs, which is characterized with the parameter αν. We assume that (Xi, Yi, 0) is the center position of the ith QD, X1 = X2 = XD and L = |Y1−Y2|. thus, αν is given by
based on
[21]. We find |α| ≤ 1 and decreases with L, and |α| = 1 corresponds to L = 0. We define
, in which
,
. With the definition of the spin polarization
in the lead ν, the tunneling matrix element can be written as
and
with
.
, Nγσ is the density of states when the superconductor is the normal state, and ργ(ε) is the modified BCS density of states
. With the RSOI phase factor φR = φR1−φR2, the spin-dependent phase factor is given by ϕσ = ϕ + 2σφR. We mainly take account of the case of the symmetric coupling between two superconducting
electrodes and DQD, that is, Γγ = Γs. According to the fluctuation-dissipation theorem, the lesser self-energy can be
given as
and
, where
. Fν and Fγ are, respectively,

(19)

(20)

in which fν(ε−eVν) = fν,
, and fs(ε) are the Fermi distribution functions. By substituting Equations 15 to 20) into Equation
6, we can obtain the spin-related current as shown in Equations 9 and 10. The AR (CAR)
coefficients (
and
) and the probability of the quasiparticle tunneling (
and
) can be calculated as

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

Thus, we can investigate the quantum transport through our model system based on the
above-mentioned equations.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

LB established the physical model and the theoretical formalism. RZ and CLD carried
out the numerical calculations and the establishment of the figures. LB performed
the physical analysis and revised the manuscript. All the authors read and approved
the final manuscript.

Acknowledgements

We thank the reviewer for the useful comments. This work is supported by the National
Natural Science Foundation of China (grant no. 11104346) and the Fundamental Research
Funds for the Central Universities (grant nos. 2011QNA28 and 2010NQB26). Chen-Long
Duan gratefully acknowledges the financial support from the Research Fund for the
Doctoral Program of Higher Education of China (grant no. 20100095120003) and China
Postdoctoral Science Foundation (grant no. 20090461153).